The morphological properties of galaxies in the range 21 < I < 25 mag in the Hubble Deep Field are investigated using a quantitative classification system based on measurements of the central concentration and asymmetry of galaxian light. The class distribution of objects in the Hubble Deep Field is strongly skewed towards highly asymmetric objects, relative to distributions from both the HST Medium Deep Survey at I < 22 mag and an artificially redshifted sample of local galaxies. The steeply rising number count-magnitude relation for irregular/peculiar/merging systems at I < 22 mag reported by Glazebrook et al. continues to at least I = 25 mag. Although these peculiar systems are predominantly blue at optical wavelengths, a significant fraction also exhibit red U - B colours, which may indicate that they are at high redshift. Beyond Glazebrook et al.'s magnitude limit, the spiral counts appear to rise more steeply than high-normalization no-evolution predictions, whereas those of elliptical/S0 galaxies only slightly exceed such predictions and may turn over beyond I ∼ 24 mag. These results are compared with those from previous investigations of faint galaxy morphology with HST, and the possible implications are briefly discussed. The large fraction of peculiar/irregular/merging systems in the Hubble Deep Field suggests that by I ∼ 25 mag the conventional Hubble system no longer provides an adequate description of the morphological characteristics of a high fraction of field galaxies.